L02_Preferences_and_Utility_2014

Report
Advanced Micro Theory
Preferences and Utility
Consumer Choice
• Postulate: an unproved and indemonstrable
statement that should be taken for granted:
used as an initial premise or underlying
hypothesis in a process of reasoning
• Consumer choice postulate: People choose
from available options to maximize their wellbeing (utility).
Criticisms
• Criticisms
– What about irrational consumers?
– Can consumers make these internal calculations?
• Irrelevant. We just want to successfully predict
behavior. To do that, we assert that all
consumers behave accordingly.
• Refutation comes if theorems that derive from
this postulate are inconsistent with the data.
– That is, if behavior contradicts the implications of the
model, then the theory is wrong.
Alternatives
• We could devise a hypothesis that postulates
that consumers
– act randomly
– do what they think society wants them to do
• But all behavior would be consistent with
these assumptions, so no refutable
implications (theoretical results) are possible…
therefore, a theory based on such a
hypothesis is useless.
Consumer Choice Model
• “People choose from available options to
maximize their well-being (utility).”
– “Available options” in the model will be handled by
the budget constraint.
• The budget constraint will provide decision-makers with
MC of choices.
– “Maximizing well-being” will be incorporated into
the model via assumptions about preferences –
which will then be used to build a utility function.
• The preferences part of the model will provide decisionmakers with the MB of choices.
Modeling Preferences
• Let bundle A = (x1, y1) and B = (x2, y2) where
the goods are x and y.
Y
A
y1
y2
B
x1
x2
X
Varian’s Version
• Let bundle X = (x1, x2) and Y = (y1, y2) where
the goods are x1 and x2 .
So the goods listed on
the axes and the
quantities of each good
in the first bundle are
the using the same
notation.
X2
X
x2
y2
Y
x1
y1
X1
Varian’s Version
• He does this to be consistent with his
advanced micro text.
good 2
In that text, he uses vector
notation and eliminates the
subscripts by not noting
quantities of each good on
the axes.
X
Y
good 1
Modeling Preferences
• IMO, students have invested so much math time with X and Y on the axes,
dy
that I want to leverage that. E.g. slope =
dx
• Also, with all the derivations coming up, we will have plenty of subscripts
floating around that I hate to add an additional set with goods X1 and X2.
Y
A
y1
y2
B
x1
x2
X
Modeling Preferences
• Three choices: A
B, consum er strictly prefers bundle A to bundle B
A
B, consum er w eakly prefers bundle A to bundle B
A
B, consum er indifferent betw een bundle A and bundle B
Y
• And therefore
A
If A
B, and B
A then A
If A
B, and not A
B then A
y1
y2
B
x1
x2
B
X
B
Axioms of Preference
• Axiom: a proposition that is assumed without proof for
the sake of studying the consequences that follow from
it (dictionary.com).
– These are based on ensuring logical consistency.
• Completeness:
A
B, or B
A , or both, m eaning A
B.
– Any pair of bundles can be compared and ordered
• Reflexivity:
A
A , or A
A
– A bundle cannot be strictly preferred to an identical bundle.
• Transitivity
Let C = (x 3 ,y 3 )
If A
B, and B
C then A
C
– Not a logical imperative according to Varian, but preferences become intractable if
people cannot choose between three bundles because A B, and B C and C A
• Continuity, next page
Continuity
• Preferences must be continuous
Rules out this situation:
•
•
•
•
Y
Ub=15
B
Ua=10
Uc=20
A
The bundle with more X is always preferred. Holding
X constant, more Y is better.
B
A, C
A
But, no matter how close C gets to A, C B
The utility function in this case must be
discontinuous (i.e. there must be a vertical jump
between A and C
C
X
Goods, Bads and Neutral Goods
•
•
•
•
Goods are good (more is better)
Bads are bad, less is better
Neutrals mean nothing to the consumer
Some goods start out good, but then become
bads if you consume too much
Possible Indifference Mappings Thus Far
Characterize the Goods
Y
Y
Y
X
X
X
X
Y
Y
Y
X
X
And we have…
Y
Both are good
but become bad
Y
Y is a neutral good
Y
Y
Two bads
X
X
X
Y
Two goods
X good and Y bad
Y
X good that
becomes
bad, Y good
X
X
X
Perfect Compliments and Substitutes
Y
Perfect Compliments:
More is only better if
you have more of the
other
X
Y
Perfect Substitutes: Two
goods, indifferent to
trading off a constant
amount of Y for X
X
Well-behaved Preferences
• If we want to avoid situations where demand
curves are upward sloping or people spend all
their money on one good, then we need wellbehaved indifference curves.
• Preferences must also be
– Monotonic
– Convex
Monotonic
• Monotonic: If bundle A is identical to B, except
A has more of at lease one good, then A B
– A.K.A, nonsatiation or “more-is-better”
– Ceteris paribus, increasing the quantity of one
good creates a bundle that is strictly preferred.
– Indifference curves must be downward sloping.
– Paired with transitivity, means indifference curves
cannot cross.
Monotonic
• These still possible
Y
Y
Y
X
X
X
Indifference Curves Cannot Cross
A
C , share an indifference curve
B
C , share an indifference curve
A
Y
A
B, transitivity
B ut A
B
C
X
B, m onotonicity
Convexity
• Convexity: People prefer more balanced
bundles.
– Let A = (x1, y1) and B = (x2, y2).
– Define C = (tx1 + (1-t)x2, ty1 + (1-t)y2)
• where 0 ≤ t ≤ 1
– then C
A and C
Y
B
A
y1
C
t y1 + (1-t)y2
B
y2
x1 tx1 + (1-t)x2 x2
X
Convexity:
Indifference Curves Bound Convex Sets
• Convexity: Bundles weakly preferred to those lying
on an indifference curve bound a convex set.
– Any bundle which is a weighted average of bundles on the
indifference curve are weakly preferred to bundles lying on
the curve.
Y
Y
A
y1
A
y1
B
y2
y2
x1
x2
B
X
x1
x2
X
Convex Preferences
• These still possible
Y
Y
X
X
Strict Convexity
Indifference Curves Bound Convex Sets
• Strictly Convex Preferences:
– Any bundle which is a weighted average of bundles on the
indifference curve are strictly preferred to bundles lying on
the curve (weights 0 > t > 1).
Y
C
A, C
B
A
y1
C
ty1 + (1-t)y2
B
y2
x1 tx1 + (1-t)x2
x2
X
– Simple convexity allows for straight line segments of the
indifference curve
– Strict convexity does not, the curve must have an
increasing slope as X increases.
Convexity
• Intuition: people prefer balanced bundles of
goods to bundles with a lot of one good and
little of the other good.
U=4
Y
U=7
Along a straight line
connecting the axis,
Utilty will rise and then
fall.
U=10
U=7
U=4
X
Convexity: Intuition
• Which implies indifference curves bow towards
the origin.
U=4
Y
U=7
U=10
U=7
U=4
X
Marginal Rate of Substitution
• The change in the consumption of the good on
the Y axis necessary to maintain utility if the
consumer increases consumption of the good on
the X axis by one unit.
• MRS = the slope of the indifference curve.
dy
• Although, dx  0
, MRS is almost always
defined as the abs value of the slope.
• In this class, M R S  dy
dx
MRS = MB
• The MRS describes, at any given point along
the indifference curve, the consumer’s
willingness to give up Y for one more X.
• It is therefore the marginal willingness to pay
for X
• I.e. it is the marginal benefit of consuming X.
Digression: Cardinal Utility
•
Utilitarians believed that utility, like temperature or height, was
something that was measurable (Cardinal utility).
–
–
•
Early neoclassical economists (e.g. Marshall) still held the idea
that for an individual, utility may be a cardinal measure.
–
–
–
•
And that a unit of utility was the same for everyone so if we could find out
how to measure it, we could redistribute to maximize social welfare.
The hope of some way of measuring utils did not survive long.
Believed marginal utility was strictly decreasing.
Marshall’s demand curve was downward sloping for this reason.
He is the reason P is on the vertical axis. Diminishing willingness to pay
reflected diminishing marginal utility.
Also believed that utility was additive, consumption of one good
did not affect the MU from another (Uxy = 0).
Digression: Ordinal Utility
• Pareto (1906) first considered the idea that ordinal
utility (ordering the desirability of different choices)
might be the way to think of utility.
• Work by Edgeworth, Fischer and Slutsky advanced
the theory.
• Hicks and Allen (1934) came up with the defining
theory… that we still use today.
• Pareto, Vilfredo (1906). "Manuale di economia politics, con una
introduzione ulla scienza sociale". Societa Editrice Libraria.
• Viner, Jacob. (1925a). "The utility concept in value theory and its
critics". Journal of political economy Vol. 33, No. 4, pp. 369-387
• Hicks, John and Roy Allen. (1934). "A reconsideration of the theory of
value". Economica Vol. 1, No. 1, pp. 52-76
The Utility Function
• A utility function is simply a way to
mathematically represent preferences.
• Utility is Ordinal: The ability to order bundles
is all that matters.
– The magnitude of the difference in utility is
meaningless as the numbers reflecting utility are
arbitrary.
– No interpersonal comparisons are possible.
The Utility Function
• A function such that
A
B if and only if U (A )  U (B )
• Preference can be represented by a
continuous U=U(A) so long as preferences are
reflexive, complete, transitive, continuous
• Note, monotonacity and convexity are not
needed.
• Monotonacity is always assumed because it
makes the existence proof easier and more
intuitive.
The Utility Function
• While we need preferences to be reflexive,
complete, transitive, continuous for utility
functions to exist.
• We need monotonacity and convexity to make
them well behaved.
• By well behaved, we want unique solutions
that are not extreme and are relatively stable.
– We don’t want individuals spending all their
income on one good or slight changes in price or
income to drastically affect the optimal choice.
Revisiting Monotonacity
• As all indifference curves are strictly
downward sloping, they will all cross a 45 deg
line.
y
d
x
Revisiting Monotonacity
• Monotonacity, Weak and Strong
– We will assume strong, so no thick indifference
curves!
U(d)
Weak Monotonacity
U(d)
d
Strong Monotonacity
d
Establishing Monotonocity
• We need to demonstrate that the indifference
curve is downward sloping.
– Say U0 = U(x, y)
– Solve for y = Y(x, U0), making the implicit function,
U = U(x, y), explicit.
– Calculate dy/dx
Example
• Say we have U = x2*y
– Solve to get:
•
•
•
•
y = U/x2
dy/dx = -2U/x3
Also, U = x2*y,
So dy/dx = -2y/x
– For all U and all x > 0, dy/dx < 0 and nonsatiation
holds.
• However, it may not be possible to solve for Y
explicitly (e.g. U = 6y5 – 3xy + 7x3)
dy/dx via Implicit Differentiation
•
We start with an implicit function (identity) defining an
indifference curve. To hold when x changes, y must
change too.
U 0  U  x, y(x )
dU 0

dU  x , y ( x ) 
dx
dx
dU 0

 U  x , y ( x )  dx
x
dx
0


U  x , y ( x )  dy
y
 U  x , y ( x )  dy
y
dx
U  x , y ( x ) 
x

dx
U  x , y ( x )  dy
y

dx
dx
U  x , y ( x ) 
x
U  x , y ( x ) 
dy
dx

U (x, y)
U
x
 x
 x
U  x , y ( x ) 
U y (x, y)
Uy
y
https://www.khanacademy.org/math/calculus/differentialcalculus/implicit_differentiation/v/implicit-differentiation-1
Example
• Start with U = x2*y
dy

UX
dx
UY
– And
U X  2 xy
Uy  x
dy
dx

2
2 xy
x
2

2y
x
– So monotonacity holds as -2y/x < 0 for all x,y >0
Intermediate Micro Version
• Take the total differential of U = U(x, y)
dU  U X  dx  U Y  dy
0  U X dx  U Y dy
U Y dy  U X dx
dy
dx

UX
UY
• This derivation requires dividing by dx, which
bothers some people, but everyone teaches it this
way (e.g. Chiang and Wainwright, p. 375)
Transformations
• It sometimes makes life computationally easier to
transform a utility function.
• Start with utility function that is well behaved
(satisfies all the axioms of preference).
• We can transform that function with no loss of
information so long as the relationship between
any bundles A and B is unchanged.
Positive Monotonic
Transformations
• Two functions with identical ordinal properties
are called Positive Monotonic Transformations
of one another.
• Both functions will create identically SHAPED
indifference curves (although the utility value
associated with each curve will differ).
Order preserving transformations
• U = U(x,y), original utility function
• UT = UT (U(x,y)), transformation function
• UT = UT (U(x,y)) is a positive monotonic transformation if UT ‘(U) >0
for all U.
U  x y
2
U e
2
 xy 
and U
and U
T
T
 U 
1
,
2
dU
T
dU
 ln U ,
dU
T
1

U  ln x  ln y a n d U
e ,
U
 0 , th e n U
1
2U
dU
T
1

T
 xy
2
 0 , th e n U
T
 xy
U
dU
T
e
U
 0 , th e n U
T
 xy
dU
U  xy a n d U
T
  U,
dU
T
 1  0
dU
U  xy a n d U
T
 10 U  U ,
2
dU
T
dU
 10  2 U  0 fo r U  5
Convexity
• That is, MRS diminishes as x increase and y
decreases. It is about curvature.
This
Y
Not this
Y
MRS = 5
MRS = 5
MRS = 2
MRS = 2
MRS = 5
MRS = 1
X
MRS = 2
X
Digression on indifference curves.
Indifference curves are often thought of as
level curves projected onto the base plane
U=U(x, y)
U
Y
This utility function
is strictly concave
as drawn
X
Indifference Curves are Level
Curves
• Level Curves
– Are a slice of the utility function at some U = U0
– Even if the utility function is not concave (as drawn
above), but only strictly quasi concave, these level curves
bound convex sets
– Convex sets and level curves
• Any line connecting two points on the same level curve lies
within the set
• So bundles with more balance than two bundles lying on an
indifference curve will provide more utility (the utility function
will be “above” any line connecting two points on an
indifference curve.
• Convexity
Convexity
– DOES NOT IN ANY WAY indicate that the utility
function is convex as opposed to concave.
– Convex functions and convex sets are two different
concepts.
• “Strictly”
– Strictly quasi–concave utility function means the
utility function has no flat spots and it’s level sets
are strictly convex
– Strictly convex level curves means the indifference
curves have no straight line segments
– “Strictly” required to ensure just one optimum
Convexity of Preferences Implies Indifference
Curves Bound Convex Sets
This will hold if the
utility function is
Strictly Quasi
Concave
U=U(x, y)
U
Y
Utility Function
Not Concave,
but Strictly Quasi
Concave as the level
sets bound convex
sets
Any point on one of these
dotted lines (exclusive
of end points), provides
more utility than the end
points
X
Convexity
• Convexity of preferences will hold if:
– dy/dx increases along all indifference curves (it
gets less negative, closer to zero)
– That is, either:
• The level sets are strictly convex
• The utility function is strictly quasi-concave
Convexity (level curves)
• dy/dx increases along all indifference curves
• We can use the explicit equation for an
indifference curve, y=Y(x, U0) and find
2
d y
dx
0
2
to demonstrate convexity.
• That is, while negative, the slope is getting
larger as x increases (closer to zero).
U U 0
Alternatively (level curves)
• As above, starting with U(x,y)=U0,
dy
dx
 M RS  
U X ( x, y )
, assum ing M R S =
U Y ( x, y )
• So convexity if
 U X ( x, y ) 
d 

2
U
(
x
,
y
)
d y

Y


0
2
dx
dx
dy
dx
Convexity (level curves)
• And, that is
 U X ( x, y ) 
d 

2
2
2
2
U
U
U

U
U

U
U yy
U
(
x
,
y
)
d y
xy
x
y
y
xx
x

Y



0
2
3
dx
dx
Uy
*Note that Uxx and Uyy need not be negative and Uy3>0
•
What of:
–
–
–
–
–
Ux > 0, monotonacity, nonsatiation
Uy > 0, monotonacity, nonsatiation
Uxx, ?
Uyy, ?
Uxy, ?
Diminishing MU vs
Diminishing MRS
• Both involve the idea of satiation. That is, the
more you consume, the less you value added
consumption.
• DMU: Consumption of other goods irrelevant
• DMRS: The value of consuming additional units
of one good along an indifference curve falls
because you are necessarily consuming less of
other goods.
Strict Quasi-Convexity (utility function)
• Convexity of preferences will hold if the utility
function is strictly quasi-concave
– A function is strictly quasi-concave if its bordered
Hessian
0
f
f
– is negative definite
H 
0
fx
fx
f xx
 0 and
x
y
H  fx
f xx
f xy
fy
f yx
f yy
0
fx
fy
H  fx
f xx
f xy  0
fy
f yx
f yy
Negative Definite (utility function)
• So the utility function is strictly quasi-concave if
– 1. –UxUx < 0
– 2. 2UxUyUxy-Uy2Uxx -Ux2Uyy > 0
• Related to the level curve result:
– Remembering that a convex level set comes from this
dx
2
( 2U xU yU xy  U y U xx  U x U y y )
2
2
d y

Uy
3
2
0
– We can see that strict convexity of the level set and strict quasi-concavity of
the function are related, and each is sufficient to demonstrate that both are
true.
With all Six Axioms/Assumptions
Y
A
B
C
U (A )  U (B )  U (C )
A
B
C
U(A)
U(B)
U(C)
X
Some Utility Functions and their
Properties
• Homotheticity of Preferences
• Elasticity of Substitution
• Functional Forms
– CES
– Cobb-Douglas
– Perfect Substitutes
– Perfect Compliments
Homothetic Preferences
• The MRS depends only on the ratio of goods
consumed.
• So any MRS that can be reduced so that x and y only
appear as the ratio (x/y) or (y/x) are considered
homothetic.
• Changes in income lead to equal percent changes in
consumption (income elasticity = 1 for all goods).
Elasticity of Substitution, 
• What is the % change in the ratio of y*/x*
when there is a 1% change in MRSxy?
Y
y*/x* = slope of
MRS at x*, y* = slope
of tangent line
y*
(0,0)
x*
X
Elasticity of Substitution, 

%  ( y * / x*)
%  (M R S)
Change in y/x
all different
Y
% change in MRS
from the slope of
the original tangent
line to each of
these is the same y*
x*
X
Elasticity of Substitution, 
• As an elasticity, it is true that

%  ( y * / x*)
%  (M R S)

d ( y * / x*)
d (M R S)

(M R S)
( y * / x*)

d ln( y * / x*)
d ln(M R S)
• And, MRS = Ux/Uy

 U x  x *, y *  

d
 U  x *, y *  

 y


, or
 y*
U x  x *, y *  



x
*


U y  x *, y *  
y*

x*

d


Evaluated
at (x*, y*)

 y*
d ln 

x
*


 U x  x *, y *  
d ln 
 U  x *, y *  
 y

And another substitution
• And at utility maximizing x*and y*, MRS = px/py, so:

%  ( y * / x*)
%  (M R S)

%  ( y * / x*)
%  (p x / p y )
• Which means, elasticity of substitution can be defined as
either of these:


d


d


p 
x
y*

  p 
x*  y 

, or
px   y * 
  x * 

py  

 y*
d ln 

 x*
p 
d ln  x 
p 
 y 
• Which has some real economic meaning. It is a measure of
the magnitude of the substitution effect.
Utility Functions
•
•
•
•
CES
Cobb-Douglas Utility
Perfect Substitutes
Perfect Compliments
CES, Constant Elasticity of Substitution

•


 
U  A   n x 
 n 1

N
CES utility:
N
w here A > 0;  n  0;
•
n
 1; ρ< 1; ρ  0;  > 0
n 1
Generally, this is simplified


U  A  x   y 




And often to
1


U    x   y  
w here A  1,  = 1
0< ρ  1

U  A  x   y  

w here A > 0; ρ  1; ρ  0;  > 0
w here A > 0; ρ  -1; ρ  0;  > 0
•

or
or

U x  y

w h ere A  1,  = 
0 < ρ  1
U 
x



y


w here A  1
 =  ,   0,   1
CES, Constant Elasticity of Substitution
•
Start with this form and find Ux:


U  A   x   y 
Ux  
A
 x
 

y

1
w h ere A > 0 ; ρ  -1 ; ρ  0 ;  = 1





1

1
     x
  1
 (   1)
Ux x
  1
M u ltip ly b y
A   x
A
A
Ux x
  1
A

y






1 
A

 x     y   



(   1)



CES, Constant Elasticity of Substitution
•
Transform the original utility function
U 
T ransform U :  
 A
U 
 
 A
1 



    x   y 

1 


   x   y 
U 
S ubstitute  
 A
Ux x
  1
A
1 
in to U x   x
A

U 
 
 A
1 
1




1 
  1 


  
1 
Y ielding

  1
A
1 
A

 x     y   



(   1)

CES, Constant Elasticity of Substitution
•
Simplify
Ux x
  1
A
1 

A
U 
 
 A
1 
A nd sim plify
Ux x
Ux 
 (   1)

A

S o, U x 
U
1 
A

U
x

1 
(  1)
 U 

 
A  x 
1 
1 
, and sim ilarly, U y 
 U 

 
A  y 
CES: MRS and σ
•
With Ux and Uy we can define MRS and σ:
MRS =
Ux
Uy

d

 

d


*
y 
* 
x 
px 

p y 







px 

p y 
*
y 
* 
x 
CES: MRS
 U 
MRS =
Ux
Uy
1 
 
A  x 


1 
 U 

 
A  y 

  y
1 
 
 x
Homothetic!
CES: σ

d

 

d


*
y 
* 
x 
px 

p y 







px 

p y 
*
y 
* 
x 
1 
  y 
 * 
  x 
*
*
*
U tility m ax requires x and y such that:
1
 y    p x  1 
S o, at U -m ax:  *    



 x    py 
*

px
py
CES: σ
•
Split it up:

d

 

d


y 
* 
x 
px 

p y 
*







 y*  
d  *  
 x  
 

 p  
x
d
 p  
 y 
px 

p y 
*
y 
* 
x 
px 

p y 
*
y 
* 
x 
S plit this into tw o parts, first deal w ith the derivative portion.

d


d



d


d


*
y 
* 
x 
1

px  1  

p y 
1
  p  1 
x
 

 p 
y 

*
y 
1
* 
1 
x 
1  

 
px  1     

p y 
1
 
 
 
1
 p  1 
x


 p 
 y 
1
1
, and
  p  1 
x

 

*
 p 
x
y 

y
*
CES: σ
•
And the other part:







px 

p y 
*
y 
* 
x 

px 


p y 


*
y 

* 
x   


px 

p y 

d

 

d








y 
* 
x 
px 

p y 
*
1
p x  1 

p y 
 
 
 

 y*  
d  *  
 x  
 

 p  
x
d
 p  
 y 
1
1 
 p 
x


 p 
 y 
1
1
1 
px 

p y 
*
y 
* 
x 
1
, and
  p  1 
x

 

*
 p 
x
y 

y
*
CES, Constant Elasticity of Substitution
•
Bring the parts back together:
 y*   px 
1

d  *  
1    1 
 x   py 
 

=
 
 p   y*  1     
d x   * 
 p   x 
 y 
•
 p  1 
x


 p 
 y 
1
 
 
 

1
1 
 p 
x


 p 
 y 
1
1
1 
Yielding
1
   1 
 
 
1    
1
 
•
1
1
1 

1
1 
1
 p  1 
x


 p 
 y 
1 1
1
1 
, rem em b er, ρ  -1
Means along a CES indifference curve, σ is
constant… well named.
CES: The Mother of All Utility Functions
 
1
1 
,   -1,   0
A s    ,   0, perfect com plim ents
A s   0,   1, C obb-D ouglas
A s    1,    , perfect substitutes
Simpler CES
• If we go with this simpler CES:
U 
x



y


w e g et
M RS =
Ux
U

y
x
y
 1
 1
w h ere   0,   1
 y*   px 

d  *  
p
 x   y 
R em em ber,  

 p   y* 
d x   *
 p   x 
 y 
*
*
U tility m ax requires x and y such that:
x
y
1
 y*   px 
S o, at U -m ax:  *   

 p 
x

  y 
1 
 1
 1

px
py
Simpler CES
• Which reduces to
 p 
x


1    p y 
1
 
 p 
x


 py 
 
1
1
 1

 1 
 1

 1 

 1


 1


1
1
, here,   1,   0
A nd this tim e
A s     ,   0, perfect com plim ents
A s   0,   1, C obb-D ouglas
A s   1,    , perfect substitutes
Cobb-Douglas
• Cobb-Douglas: U = Axαyβ
MRS 
y
x
Homothetic
• When 0<α<1 and 0<β<1 and α+β=1
– α, share of income spent on x
– β, share of income spent on y
• To get this, transform: UT = (U)(1/(α+β))
– UT = (xαyβ) 1/(α+β)
– UT = (x(α/(α+β))y(β/(α+β)))
– (α/α+β) + (β/α+β) = 1
• But how is Cobb-Douglas derived from CES?
CES to Cobb-Douglas
•
First, a digression:
ˆ
L'H opital's
rule: lim
a
m( )
n( )
 lim
a
m (  )
n (  )
if lim m (  )  0 and lim n (  )  0
0
•
0
So now we need to split the Utility function
into a ratio of two functions of ρ.
CES to Cobb-Douglas
•
Utility function:
U (  )  A   x
•

y




1

w here A > 0; ρ  -1; ρ  0;     1
A monotonic transformation.
T
U = ln
U
A
U =
T
T
U =

1


 ln   x   y 

1


 ln   x   y 



 ln   x   y 



m (  )   ln   x   y 
n( )  
CES to Cobb-Douglas
•
And with this
T
U =


 ln   x   y 



m (  )   ln   x   y 
n( )  
•
Back to L’Hopital’s Rule.
L 'H oˆ p ital's ru le: lim
0
S o : lim
0
 ln   x


m( )
n( )
y

 lim
0
m (  )
n (  )
d
  ln   x     y    



d 

 lim
0
d
d
 
CES to Cobb-Douglas
•
Derivatives and the limits
d
  ln   x     y    




    x ln x   y ln y 
d 
lim
= lim
0
0
d
 x     y   


 
d
lim
    x
 x

0
lim
0
lim
0

m (  )
n (  )
m (  )
n (  )
ln x   y

y


ln y 


  ln x   ln y
=  ln x   ln y  ln  x y

= ln  x y



 U ( ) 


S o, lim ln 

ln
x
y


0
 A 

B ut w e need lim U (  ), not lim U
0
0
T


CES, Constant Elasticity of Substitution
•
A little rearranging:
 U ( ) 


S o, lim l n 
 ln  x y

0
 A 
e
U ( ) 
l im ln 

 A 
0
Ae
Ae
•
=e
U ( ) 
lim ln 

 A 
0


ln x y
=Ae
U ( ) 
lim ln 

0
 A 
Yielding
li m U (  ) = A e




ln x y

 A lim e
0


ln x y



U ( ) 
ln 

 A 
 U ( ) 
 lim A 
m U ( )
  li
0


0
 A 

0

li m U (  )  A x y

0
•
And so long as α+β=1, CES becomes Cobb-Douglas as
ρ→0
Perfect Substitutes
• Start with a version of CES


U   x   y , and  
1
1
, w here,   1,   0
lim U   x   y
 1
MRS = α/β, does not depend on x or y, or y/x
No diminishing MRS, not homothetic
Perfect Substitutes
• Example: If Ozarka Water (12oz) and Dasani Water
are (24oz), then
U = αO+ βD, β=2α
U = αO+ 2αD
• MRSOD = ½
• Willing to give up ½ a Dasani for 1 Ozarka
D
5
10
O
Perfect Compliments
• U = min(αx, βy), where α, β >0
– Utility = the smaller of αx or βy
– Vertex where αx = βy, or where y/x = α/β
• Example: You always eat 3 T of Nutella with 2
slices of bread.
N on vert. axis, vertex
U = min(3B, 2N)
When B = 2 and N = 3, U = 6
When B = 4 and N = 3, U = 6
When B = 2 and N = 6, U = 6
When B = 4 and N = 6, U = 12
N
where
N/B = 3/2, or N = 3/2 B
Bread
Neoclassical Behavioral Assertion
• Consumers endeavor to maximize
U  U ( xi )
where U(xi) represents the consumer’s own
subjective evaluation of derived from the
consumption of goods and services, xi.
• Under scarcity, consumers must choose among a
limited set of bundles described by the budget
constraint
 px M
where pi are the prices of xi goods and services and
M is consumer income.
i
i
The Hypothesis
• All consumers endeavor to maximize
U  U ( xi )
subject to the budget constraint

pi xi  M
• So we have beaten Utility to death.
• Next week we add in the constraint and
solve for the optimal x* and y*
Spare: MRS diminishing?
• U = x+xy+y
o MRS = Ux/Uy = (1+y)/(1+x)
o Is 2UxUyUxy-Uy2Uxx -Ux2Uyy > 0 ?
o 2(1+y)(1+x)*1-(1+x)2*0-(1+y)2*0
o 2(1+y)(1+x) > 0
• U = x2y2
o MRS = 2xy2/2x2y = y/x
o Is 2UxUyUxy-Uy2Uxx -Ux2Uyy>0 ?
o 2(2xy2)(2x2y)(4xy)-(4x4y2)(2y2)-(4x2y4)(2x2) > 0
o 32x4y4 - 16x4y4 >0
o 16x4y4 > 0
Diminishing MRS
 U x  x, y  

dx
dy 
d
U xy
Uy  Ux
 U  x , y      U xx
dx
dx 
y




2
dx
Uy
N ote:
dy

dx
dx
dy  

 U yy
 U yx

dx
dx  

Ux
Uy
 U x  x, y  
d
 U  x , y  
y



dx


Ux 
Ux 
  U xx  U xy
 U y  U x  U yx  U yy

Uy 
U y 


2
Uy
2
 U x  x, y  
Ux
d
 U xx U y  U xy U x  U yx U x  U yy
 U  x , y  
Uy
y



2
dx
Uy
M ultiply by:
Uy
Uy
 U x  x, y  
d
 U  x , y    U U 2  U U U  U U U  U U 2
y
xx
y
xy
x
y
yx
x
y
yy
x



3
dx
Uy
 U x  x, y  
d
 U  x , y   2 U U U  U U 2  U U 2
y
xy
x
y
xx
y
yy
x



0
3
dx
Uy

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